Some asymptotic results on q-binomial coefficients

We look at the asymptotic behavior of the coefficients of the $q$-binomial coefficients (or Gaussian polynomials) ${\left(\genfrac{}{}{0px}{}{a+k}{k}\right)}_q$, when $k$ is fixed. We give a number of results in this direction, some of which involve Eulerian polynomials and their generalizations.